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To perform this measurement, you would need a whirling hygrometer (sling pyschrometer) such as Elcometer 116.
Use it to establish the wet bulb and dry bulb temperature, and then calculate humidity and dewpoint temperature. You can use a conversion tables, such as ones supplied with the Elcometer 116. Alternatively, you can follow this easy procedure using Elcometer 114 Dew Point Calculator:
1. Obtain the measurement using a whirl hygrometer. For exemple, If these are wet bulb and dry bulb temperatures: 
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Look at the top half of the disc on the Elcometer 114 Dew Point Calculator, find scale 1 on top, and align the wet bulb temp. with the dry bulb temp. 
In scale 2, locate the wet bulb temperature again, and read the corresponding measurement. This is the dewpoint temperature. 
2.Secondly, looking at the bottom half of the disc, on scale 3, align the dewpoint temperature with the dry bulb temperature. 
This reveals your relative humidity in the window above: 
Related article:
Calculating dew point from relative humidity and temperature
